We consider the so-called Thomson problem which refers to finding the equilibrium distribution of a finite number of mutually repelling point charges on the surface of a sphere, but for the case where the sphere is re...We consider the so-called Thomson problem which refers to finding the equilibrium distribution of a finite number of mutually repelling point charges on the surface of a sphere, but for the case where the sphere is replaced by a spheroid or ellipsoid. To get started, we first consider the problem in two dimensions, with point charges on circles (for which the equilibrium distribution is intuitively obvious) and ellipses. We then generalize the approach to the three-dimensional case of an ellipsoid. The method we use is to begin with a random distribution of charges on the surface and allow each point charge to move tangentially to the surface due to the sum of all Coulomb forces it feels from the other charges. Deriving the proper equations of motion requires using a projection operator to project the total force on each point charge onto the tangent plane of the surface. The position vectors then evolve and find their final equilibrium distribution naturally. For the case of ellipses and ellipsoids or spheroids, we find that multiple distinct equilibria are possible for certain numbers of charges, depending on the starting conditions. We characterize these based on their total potential energies. Some of the equilibria found turn out to represent local minima in the potential energy landscape, while others represent the global minimum. We devise a method based on comparing the moment-of-inertia tensors of the final configurations to distinguish them from one another.展开更多
A quantum steering ellipsoid(QSE)is a visual characterization for bipartite qubit systems,and it is also a novel avenue for describing and detecting quantum correlations.Herein,by using a QSE,we visualize and witness ...A quantum steering ellipsoid(QSE)is a visual characterization for bipartite qubit systems,and it is also a novel avenue for describing and detecting quantum correlations.Herein,by using a QSE,we visualize and witness the first-order coherence(FOC),Bell nonlocality(BN)and purity under non-inertial frames.Also,the collective influences of the depolarizing channel and the non-coherence-generating channel(NCGC)on the FOC,BN and purity are investigated in the QSE formalism.The results reveal that the distance from the center of the QSE to the center of the Bloch sphere visualizes the FOC of a bipartite system,the lengths of the QSE semiaxis visualize the BN,and the QSE's shape and position dominate the purity of the system.One can capture the FOC,BN and purity via the shape and position of the QSE in the non-inertial frame.The depolarizing channel(the NCGC)gives rise to the shrinking and degradation(the periodical oscillation)of the QSE.One can use these traits to visually characterize and detect the FOC,BN and purity under the influence of external noise.Of particular note is that the condition for the QSE to achieve the center of the Bloch sphere cannot be influenced by the depolarizing channel and the NCGC.The characterization shows that the conditions for the disappearance of the FOC are invariant under the additional influences of the depolarizing channel and NCGC.展开更多
文摘We consider the so-called Thomson problem which refers to finding the equilibrium distribution of a finite number of mutually repelling point charges on the surface of a sphere, but for the case where the sphere is replaced by a spheroid or ellipsoid. To get started, we first consider the problem in two dimensions, with point charges on circles (for which the equilibrium distribution is intuitively obvious) and ellipses. We then generalize the approach to the three-dimensional case of an ellipsoid. The method we use is to begin with a random distribution of charges on the surface and allow each point charge to move tangentially to the surface due to the sum of all Coulomb forces it feels from the other charges. Deriving the proper equations of motion requires using a projection operator to project the total force on each point charge onto the tangent plane of the surface. The position vectors then evolve and find their final equilibrium distribution naturally. For the case of ellipses and ellipsoids or spheroids, we find that multiple distinct equilibria are possible for certain numbers of charges, depending on the starting conditions. We characterize these based on their total potential energies. Some of the equilibria found turn out to represent local minima in the potential energy landscape, while others represent the global minimum. We devise a method based on comparing the moment-of-inertia tensors of the final configurations to distinguish them from one another.
基金Project supported by the National Natural Science Foundation of China(Grant No.12175001)the Natural Science Research Key Project of the Education Department of Anhui Province of China(Grant No.KJ2021A0943)+3 种基金the Research Start-up Funding Project of High Level Talent of West Anhui University(Grant No.WGKQ2021048)an Open Project of the Key Laboratory of Functional Materials and Devices for Informatics of Anhui Higher Education Institutes(Grant No.FMDI202106)the University Synergy Innovation Program of Anhui Province(Grant No.GXXT-2021-026)the Anhui Provincial Natural Science Foundation(Grant Nos.2108085MA18 and 2008085MA20)。
文摘A quantum steering ellipsoid(QSE)is a visual characterization for bipartite qubit systems,and it is also a novel avenue for describing and detecting quantum correlations.Herein,by using a QSE,we visualize and witness the first-order coherence(FOC),Bell nonlocality(BN)and purity under non-inertial frames.Also,the collective influences of the depolarizing channel and the non-coherence-generating channel(NCGC)on the FOC,BN and purity are investigated in the QSE formalism.The results reveal that the distance from the center of the QSE to the center of the Bloch sphere visualizes the FOC of a bipartite system,the lengths of the QSE semiaxis visualize the BN,and the QSE's shape and position dominate the purity of the system.One can capture the FOC,BN and purity via the shape and position of the QSE in the non-inertial frame.The depolarizing channel(the NCGC)gives rise to the shrinking and degradation(the periodical oscillation)of the QSE.One can use these traits to visually characterize and detect the FOC,BN and purity under the influence of external noise.Of particular note is that the condition for the QSE to achieve the center of the Bloch sphere cannot be influenced by the depolarizing channel and the NCGC.The characterization shows that the conditions for the disappearance of the FOC are invariant under the additional influences of the depolarizing channel and NCGC.
